Singular perturbation theory for interacting fermions in two dimensions

We consider a system of interacting fermions in two dimensions beyond the second-order perturbation theory in the interaction. It is shown that the mass-shell singularities in the self-energy, arising already at the second order of the perturbation theory, manifest a nonperturbative effect: an interaction with the zero-sound mode. Resumming the perturbation theory for a weak, short-range interaction and accounting for a finite curvature of the fermion spectrum, we eliminate the singularities and obtain the results for the quasiparticle self-energy and the spectral function to all orders in the interaction with the zero-sound mode. A threshold for emission of zero-sound waves leads a nonmonotonic variation of the self-energy with energy (or momentum) near the mass shell. Consequently, the spectral function has a kinklike feature. We also study in detail a nonanalytic temperature dependence of the specific heat C (T) T2. It turns out that although the interaction with the collective mode results in an enhancement of the fermion self-energy, this interaction does not affect the nonanalytic term in C (T) due to a subtle cancellation between the contributions from the real and imaginary parts of the self-energy. For a short-range and weak interaction, this implies that the second-order perturbation theory suffices to determine the nonanalytic part of C (T). We also obtain a general form of the nonanalytic term in C (T), valid for the case of a generic Fermi liquid, i.e., beyond the perturbation theory.